ESTIMATION IN HIGH-DIMENSIONAL LINEAR MODELS WITH DETERMINISTIC DESIGN MATRICES

• 成果类型: Article
• 署名作者: Shao, Jun; Deng, Xinwei
• 署名单位: East China Normal University; University of Wisconsin System; University of Wisconsin Madison; Virginia Polytechnic Institute & State University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/12-AOS982
• 发表日期: 2012
• 页码: 812-831
• 关键词: nonconcave penalized likelihood; variable selection; oracle properties; feature space; Lasso; regression

摘要

Because of the advance in technologies, modem statistical studies often encounter linear models with the number of explanatory variables much larger than the sample size. Estimation and variable selection in these high-dimensional problems with deterministic design points is very different from those in the case of random covariates, due to the identifiability of the high-dimensional regression parameter vector. We show that a reasonable approach is to focus on the projection of the regression parameter vector onto the linear space generated by the design matrix. In this work, we consider the ridge regression estimator of the projection vector and propose to threshold the ridge regression estimator when the projection vector is sparse in the sense that many of its components are small. The proposed estimator has an explicit form and is easy to use in application. Asymptotic properties such as the consistency of variable selection and estimation and the convergence rate of the prediction mean squared error are established under some sparsity conditions on the projection vector. A simulation study is also conducted to examine the performance of the proposed estimator.